cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A276485 Numerator of Sum_{k=1..n} 1/k^n.

Original entry on oeis.org

1, 5, 251, 22369, 806108207, 47464376609, 774879868932307123, 248886558707571775009601, 4106541588424891370931874221019, 413520574906423083987893722912609, 7429165883912264897181708263009894640627544300697
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 05 2016

Keywords

Comments

Also numerators of zeta(n) - Hurwitz zeta(n,n+1), where zeta(s) is the Riemann zeta function and Hurwitz zeta(s,a) is the Hurwitz zeta function.
Sum_{k>=1} 1/k^n = zeta(n).

Examples

			1, 5/4, 251/216, 22369/20736, 806108207/777600000, 47464376609/46656000000, 774879868932307123/768464444160000000, ...
a(3) = 251, because 1/1^3 + 1/2^3 + 1/3^3 = 251/216.
		

Crossrefs

Cf. A001008, A002805, A007406, A007407, A031971, A276487 (denominators).

Programs

  • Mathematica
    Table[Numerator[HarmonicNumber[n, n]], {n, 1, 11}]
  • PARI
    a(n) = numerator(sum(k=1, n, 1/k^n)); \\ Michel Marcus, Sep 06 2016

A322266 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = denominator of Sum_{j=1..n} 1/j^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 8, 36, 12, 1, 1, 16, 216, 144, 60, 1, 1, 32, 1296, 1728, 3600, 20, 1, 1, 64, 7776, 20736, 216000, 3600, 140, 1, 1, 128, 46656, 248832, 12960000, 24000, 176400, 280, 1, 1, 256, 279936, 2985984, 777600000, 12960000, 8232000, 705600, 2520, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 01 2018

Keywords

Examples

			Square array begins:
  1,       1,          1,              1,                  1,  ...
  2,     3/2,        5/4,            9/8,              17/16,  ...
  3,    11/6,      49/36,        251/216,          1393/1296,  ...
  4,   25/12,    205/144,      2035/1728,        22369/20736,  ...
  5,  137/60,  5269/3600,  256103/216000,  14001361/12960000,  ...
		

Crossrefs

Numerators are in A322265.

Programs

  • Mathematica
    Table[Function[k, Denominator[Sum[1/j^k, {j, 1, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
    Table[Function[k, Denominator[HarmonicNumber[n, k]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten
    Table[Function[k, Denominator[SeriesCoefficient[PolyLog[k, x]/(1 - x), {x, 0, n}]]][i - n], {i, 0, 10}, {n, 1, i}] // Flatten

Formula

G.f. of column k: PolyLog(k,x)/(1 - x), where PolyLog() is the polylogarithm function (for rationals Sum_{j=1..n} 1/j^k).
Showing 1-2 of 2 results.